The discrete KP hierarchy and the negative power series on the complex plane
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The discrete KP hierarchy and the negative power series on the complex plane Nancy López Reyes · Raúl Felipe · Tovias Castro Polo
Received: 14 September 2012 / Revised: 18 February 2013 / Accepted: 1 March 2013 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013
Abstract In this article, we have connected the control theory of linear infinite-dimensional systems and the discrete KP integrable system. We show how the space of negative power series of z on {z ∈ C : |z| > 1} can be parametrized by means of an integrable system. This study is a kind of extension to infinite-dimensional case of some results discussed in Felipe and López-Reyes (Discret Dyn Nat Soc 2008, 2008). Keywords system
Discrete KP hierarchy · Integrable system · State linear infinite-dimensional
Mathematics Subject Classification (2000)
37K10 · 93B27
1 Introduction During the past decades, several authors have studied an interesting relation between the dynamical system control theory and the integrable systems. For example see Felipe and López-Reyes (2008) and Nakamura (1991), which deal with problems related to those discussed here, are examples of these researches. It is well known that a nonlinear (completely) integrable system is always related to some type of group factorization, and also that it describes isospectral deformations of some
Communicated by José Eduardo Souza de Cursi. N. López Reyes (B) · T. Castro Polo Facultad de Ciencias Exactas y Naturales, Instituto de Matemáticas, Universidad de Antioquia, Medellin1226, Colombia e-mail: [email protected] R. Felipe Centro de Investigación en Matemáticas (CIMAT), Callejón Jalisco s/n Mineral de Valenciana, Guanajuato, Gto., México e-mail: [email protected]
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linear operator. On the other hand, a “hierarchy” is an integrable system which indicates a differential equation system that describes set of commutative flows on the solution spaces of original equation. In this paper, we connect the control theory of linear dynamical infinite-dimensional systems (Bensoussan et al. 2007; Curtain and Zwart 1995) with the nonlinear integrable system indicated by the discrete KP hierarchy (Felipe and Ongay 2001) which describes isospectral deformations of the backward shift operator (Fuhrmann 1996). In this way, we extent some results of our earlier work (Felipe and López-Reyes 2008) where we have established a relation between a finite-dimensional version of the KP hierarchy and the control theory of linear systems. We will discuss a state linear system with semigroup T (τ ) = eτ . We have proved that this system is approximately controllable and approximately observable. Further, using the group factorization of the discrete KP hierarchy (Felipe and Ongay 2001) it have been shown that these dual properties are kept in the parametric state linear system with semigroup T (τ ) = eτ L(t) where L(t) is a solution (isospectral deformation of ) of discrete KP hierarchy. Also, via realization theory, it is showed that the space of compl
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